Numerical analysis of the continuum formulation for the initial evolution of mixing layers with particles

Numerical analysis of the standard continuum description of a dilute dispersed phase as applied to a laminar, particle-laden, mixing layer during its initial evolution has been performed. The flow has been previously analyzed under the framework of linear stability analysis where both the continuous and the dispersed phases are considered as continua. Earlier studies had neglected the closure terms resulting from the averaging of the nonlinear transport term involved in the derivation of the dispersed-phase momentum equations. In this work, Lagrangian particle tracking was coupled to an incompressible Navier–Stokes solver to directly estimate the closure terms (referred to as the averaging-stress terms) and compare them to the other terms balancing the dispersed-phase continuum equations. Calculations were performed for particle Stokes numbers of 1, 10, and 50, and for a mass loading of one. Dispersed-phase flow quantities such as the number density and velocity were determined by averaging the data in the spanwise direction. A parametric study of the influence of the number of particles, for Stokes number of one, showed that an improved approximation to a continuum can be obtained by increasing the number of particles. Examining the momentum balance in detail revealed that the main contributors were the time-derivative, convective, and the interfacial force terms. The averaging stress was at least two orders of magnitude smaller for all the Stokes numbers studied. However, the averaging stress, though negligible in magnitude, showed a deterministic variation in the center of the mixing layer. The results lend support to the currently used continuum equations for analyzing the stability of laminar, particle-laden mixing layers, and possibly other free-shear flows such as jet and wake flows.